Problem: Sakura speaks $150$ words per minute on average in Hungarian, and $190$ words per minute on average in Polish. She once gave cooking instructions in Hungarian, followed by cleaning instructions in Polish. Sakura spent $5$ minutes total giving both instructions, and spoke $270$ more words in Polish than in Hungarian. How long did Sakura speak in Hungarian, and how long did she speak in Polish? Sakura spoke for
Explanation: Let $x$ represent the time (in minutes) Sakura spoke in Hungarian and let $y$ represent the time (in minutes) she spoke in Polish. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that Sakura speaks an average of $\textit{150}$ words per minute in Hungarian, an average of $\textit{190}$ words per minute in Polish, and spoke $\textit{270}$ more words in Polish than in Hungarian. How can we model this sentence algebraically? The total number words Sakura spoke in Hungarian can be modeled by $150x$, and the total number words she spoke in Polish can be modeled by $190y$. Since she spoke $270$ more words in Polish than in Hungarian, we get the following equation: $150x+270=190y$ We can rewrite this equation in standard form as such: $150x-190y=-270$. We are also given that Sakura gave instructions for a total of $\textit{5}$ minutes. This can be expressed as: $x + y =5$ Let's rewrite this equation so that it's solved for $x$ : $ x = 5-y$ Now that we have a system of two equations, we can go ahead and solve it! Let's substitute $ x={5- y}$ into the first equation: $\begin{aligned}150 x-190 y &= -270\\\\ 150 \cdot ({5-y})-190y&=-270\\\\ 750-150y-190y&=-270\\\\ -340y&=-1020\\\\ y&=3\end{aligned}$ Now we can substitute $y = 3$ into $x=5- y$ and find that $x=2$. Recall that $x$ denotes the time Sakura spoke in Hungarian and $y$ denotes the time she spoke in Polish. Therefore, Sakura spoke for $\textit{2}$ minutes in Hungarian and for $\textit{3}$ minutes in Polish.